STABILIZER TOPOLOGY OF HOOPS

Borzooei, R.A.; Aaly Kologani, M.

STABILIZER TOPOLOGY OF HOOPS

Document Type : Research Paper

Authors

R.A. Borzooei ¹
M. Aaly Kologani ²

¹ Shahid Beheshti University

² Payamenour University, Tehran

Abstract

In this paper, we introduce the concepts of right, left and product stabilizers on hoops and study some properties and the relation between them. And we try to find that how they can be equal and investigate that under what condition they can be filter, implicative filter, fantastic and positive implicative filter. Also, we prove that right and product stabilizers are filters and if they are proper, then they are prime filters. Then by using the right stabilizers produce a basis for a topology on hoops. We show that the generated topology by this basis is Baire, connected, locally connected and separable and we investigate the other properties of this topology. Also, by the similar way, we introduce the right, left and product stabilizers on quotient hoops and introduce the quotient topology that is generated by them and investigate that under what condition this topology is Hausdorff space, $T_{0}$ or $T_{1}$ spaces.

Keywords

References

[1] P. Aglian´o, I. M. A. Ferreirim, F. Montagna, Basic hoops: an algebraic study of continuous t-norm, draft, (2000).
[2] B. Bosbach, Komplement¨are Halbgruppen. Axiomatik und Arithmetik, Fundamenta Mathematicae, Vol. 64 (1969),
257-287.
[3] B. Bosbach, Komplement¨are Halbgruppen. Kongruenzen and Quotienten, Fundamenta Mathematicae, Vol. 69
(1970), 1-14.
[4] M. Botur, A. Dvureˇcenskij, T. Kowalski, On normal-valued basic pseudo-hoop, Soft Comput, Vol. 16, (2012), 635-
644.
[5] N. Bourbaki, Topologie G´en´erale, Springer Berlin Heidelberg, (2007).
[6] J. R. B¨uchi, T. M. Owens, Complemented monoids and hoops, unpublished manuscript, (1975).
[7] G. Georgescu, L. Leustean, V. Preoteasa, Pseudo-hoops, Journal of Multiple-Valued logic and Soft Computing, Vol.
11. No 1-2, (2005), 153-184.
[8] P. H´ajek, Metamathematics of fuzzy logic, Springer, Vol. 4. (1998).
[9] K. D. Joshi, Introduction to general topology, New Age International Publisher, India, (1983).
[10] Y. B. Jun, H. S. Kim, Uniform structures in positive implication algebras, Intern. Math. J. Vol. 2, No 2, (2002),
215-219.
[11] Y. B. Jun, E. H. Roh, On uniformities of BCK-algebras, Commun. Korean Math. Soc. Vol. 10, No 1, (1995), 11-14.
[12] M. Kondo, Some types of filters in hoops, Multiple-Valued Logic (ISMVL), (2011), 41st IEEE International Symposium
on. IEEE, 50-53.
[13] J. R. Munkres, Topology a first course, Prentice-Hall, (1975).

[14] B. T. Sims, Fundamentals of Topology, Macmillan Publishing Co., Inc., New York, (1976).
[15] D. S. Yoon, H. S. Kim, Uniform structures in BCI-algebras, Commun. Korean Math. Soc. Vol. 17, No 3, (2002),
403-408.